4 1. Jacobi operators Furthermore, £Q(I,M) denotes the set of sequences with only finitely many values being nonzero and -^_(Z,M) denotes the set of sequences in ^(Z, M) which are £p near ±00, respectively (i.e., sequences whose restriction to ^(±N, M) belongs to £ P (±N,M). Here N denotes the set of positive integers). Note that, according to our definition, we have (1.3) £0(I, M) C £P(I, M) C JP(I, M) C e°°(I, M), p q, with equality holding if and only if / is finite (assuming dimM 0). In addition, if M is a (separable) Hilbert space with scalar product (.,..)M, then the same is true for £2(I, M) with scalar product and norm defined by n£l (1-4) 11/1 1 = II/II2 = v ^ 7 . f,gef(I,M). For what follows we will choose / = Z for simplicity. However, straightforward modifications can be made to accommodate the general case / C Z. During most of our investigations we will be concerned with difference expres- sions, that is, endomorphisms of £(Z) R: t(Z) -+ £(Z) U '5) f » Rf (we reserve the name difference operator for difference expressions defined on a subset of £2(Z)). Any difference expression R is uniquely determined by its corre- sponding matrix representation (1.6) ( # ( r a , n ) ) m n e Z , R(m,ri) = (RSn)(m) = (8m,R8n), where (1.7) 6n{m) = 6min = i 1 n = m is the canonical basis of £(Z). The order of R is the smallest nonnegative integer N = N++N_ such that R{m, n) = 0 for all m, n with n m N+ and m n 7V_. If no such number exists, the order is infinite. We call R symmetric (resp. skew-symmetric) if R(m, n) = R(n, m) (resp. R{m,n) = —R(n,m)). Maybe the simplest examples for a difference expression are the shift expres- sions (1.8) (S±f)(n)=f(n±l). They are of particular importance due to the fact that their powers form a basis for the space of all difference expressions (viewed as a module over the ring £{T)). Indeed, we have (1.9) i? = £ ( . , . + fc)(S+)fe, ( 5 ± ) " 1 = 5 T . kez Here i?(.,. + k) denotes the multiplication expression with the sequence (R(n,n + k))n£Zi that is, i?(.,. + k) : f(n) i-» R(n,n + k)f(n). In order to simplify notation we agree to use the short cuts (1.10) f±=S±f, f++=S+S+f, etc.,
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